"Symplectic leaves"의 두 판 사이의 차이

2012년 6월 9일 (토) 15:58 판

[[symplectic geometry|]]

‹william›The symplectic leaves are equivalence relations \(x \tilde y\) if and only if \(x\) can be connected to \(y\) be a piece-wise Hamiltonian path 27/02/201123:12:31‹william›\(x \sim y\)27/02/201123:14:18‹william›Let \(D\) be a degenerate distribution27/02/201123:14:49‹william›this means that for every point \(x \in M\), \(D_x\) is a subset of \(T_x M\)27/02/201123:14:58‹william›subset = subspace27/02/201123:15:18‹william›distribution normally means that \(D_x\) is constant rank27/02/201123:15:25‹william›and \(D_x\) is spanned by vector fields27/02/201123:15:58‹william›which means that for every \(x\) there is vector fields \(X_1,\ldots,X_r\) locally defined around \(x\) such that \(X_1(x),\ldots,X_r(x)\) span \(D_x\)27/02/201123:16:19‹william›and \(X_1(y),\ldots,X_r(y)\) lie in \(D_y\) for all \(y\) where they are defined27/02/201123:18:02‹william›a foliation of \(D\) is an immersed manifold \(A\) of \(M\) with \(TA = D\)27/02/201123:19:10‹william›Let \(M^{dis}\) be the manifold with underlying set \(M\) and the discrete topology27/02/201123:20:09‹william›\(M^{dis}\) is an immersed manifold for \(D = M \times 0\)27/02/201123:20:36‹william›\(M = \R^2\)27/02/201123:20:46‹william›\(D = \R^2 \times \R\)27/02/201123:24:17‹william›the foliation is the map \(\bigcup_{\R} \R \arr \R^2\)27/02/201123:24:25‹william›the foliation is the map \(\bigcup_{\R} \R \rightarrow \R^2\)

@@ 1번째 줄: / 1번째 줄: @@
-[[symplectic geometry|sympletic geometry]]
+[[symplectic geometry|]]
 ‹william›The symplectic leaves are equivalence relations <math>x \tilde y</math> if and only if <math>x</math> can be connected to <math>y</math> be a piece-wise Hamiltonian path
@@ 12번째 줄: / 10번째 줄: @@
 <h5>3/6/2011</h5>
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-/03/201122:37:11* '''chlee''' joins wschlsunjan30201106/03/201122:36:51* william joins wschlsunjan30201106/03/201122:39:37‹william›M : symplectic manfold of dimension <math>2n</math>06/03/201122:39:48‹william›<math>\omega</math> symplectic form06/03/201122:40:11‹william›Any function <math>H</math> can be regarded as a Hamiltonian, gives rise to a Hamiltonian vector field <math>X_H</math>06/03/201122:40:23‹william›Then we get the Hamiltonian system <math>x'(t) = X_H(x(t))</math>06/03/201122:40:54‹william›The key property is that the flows of <math>X_H</math> preserve the symplectic form06/03/201122:41:08* '''chlee''' quit (timeout)06/03/201122:41:16‹william›still there?06/03/201122:42:45‹william›A solution to the Hamiltonian equation is a integral curve <math>x(t)</math>06/03/201122:42:54‹william›which is supposed to be an equation of motion06/03/201122:43:18‹william›A flow is <math>\phi_t</math>06/03/201122:43:35‹william›Let's say that is <math>M</math> is compact06/03/201122:43:59‹william›A flow is a set of diffeomorphisms of <math>M</math> parameterized by <math>t</math>, get <math>\phi_t</math>06/03/201122:44:30‹william›if you pick a point <math>x \in M</math>, then the curve <math>\phi_t(x)</math> is an integral curve going through <math>x</math> at time <math>t=0</math>06/03/201122:44:42‹william›(<math>\phi_t</math> is a flow requires <math>\phi_0</math> to be the identity)06/03/201122:45:30‹william›Flows preserving the symplectic form means that <math>\phi_t^* \omega = \omega</math> for all <math>t</math>06/03/201122:45:42‹william›<math>M</math> has a volume form <math>\omega^n</math>06/03/201122:46:19‹william›Symplectic form preserving flows are volume preserving06/03/201122:47:42‹william›We also have a Poisson bracket <math>{,}</math>06/03/201122:47:48‹william›<math>\{,\}</math>06/03/201122:48:08‹william›<math>\{f,g\}</math>06/03/201122:48:24‹william›<math>\{f,g\} = \omega(X_f,X_g)</math>06/03/201122:48:41‹william›The other thing Hamiltonian flows preserve are first integrals06/03/201122:48:53‹william›first integrals are functions <math>f</math> such that <math>\{f,H\} = 0</math>06/03/201122:49:14‹william›If <math>x</math> is a Hamiltonian curve then <math>f(x(t))</math> is constant for all time <math>t</math>06/03/201122:49:25‹william›<math>\{H,H\} = 0</math>06/03/201122:49:36‹william›<math>H</math> itself is a first integral06/03/201122:49:42‹william›example <math>N</math> configuration space06/03/201122:49:51‹william›<math>\omega</math> canonical symp. form on <math>T^* N</math>06/03/201122:50:02‹william›<math>H</math> being the kinetic plus potential energy06/03/201122:51:02‹william›More generally, if <math>f_1,\ldots,f_k</math> are first integrals, then can think of <math>f</math> as a function <math>M \arr \R^k</math>06/03/201122:51:13‹william›<math>f : M \rightarrow \R^k</math>06/03/201122:51:31‹william›And the Hamiltonian curves stay within the level sets <math>f^{-1}(c)</math>06/03/201122:51:47‹william›An integrable system is a set of <math>f_1,\ldots,f_n</math>06/03/201122:51:58‹william›which are (1) first integrals06/03/201122:52:28‹william›(2) <math>\{f_i,f_j\} = 0</math>06/03/201122:52:51‹william›(3) <math>df_1 \wedge \cdots \wedge df_n</math> is non-zero06/03/201122:53:10‹william›(3) is equivalent to saying that <math>f : M \rightarrow \R^n</math> has a regular value06/03/201122:54:19‹william›<math>M</math> has dimension <math>2n</math>06/03/201122:54:58‹william›Let's say that I have <math>f_1,\ldots,f_k</math> such that <math>\{f_i,f_j\} = 0</math>06/03/201122:55:11‹william›Then <math>k \leq n</math>, proof:06/03/201122:55:18‹william›Look at <math>X_{f_i}</math>06/03/201122:55:32‹william›<math>\omega(X_{f_i},X_{f_j}) = 0</math>06/03/201122:56:16‹william›(also assume that <math>df_1 \wedge \cdots df_k \neq 0</math>06/03/201122:56:19‹william›)06/03/201122:56:44‹william›This is equivalent to saying that <math>X_{f_1},\ldots,X_{f_k}</math> are linearly independent at some point06/03/201122:57:18‹william›Thus there is a point <math>c</math> such that the span <math>V</math> of <math>X_{f_1}(c),\ldots,X_{f_k}(c)</math>06/03/201122:57:25‹william›satisfies <math>\omega_{V} = 0</math>06/03/201122:57:34‹william›<math>\omega|_V = 0</math>06/03/201122:57:42‹william›So <math>\dim V \leq n</math>06/03/201123:00:06‹william›Say <math>\omega = \sum dp^i \wedge dq^i</math>06/03/201123:01:08‹william›---06/03/201123:01:32‹william›since we get <math>n</math> functions, this is sometimes called a completely integrable system06/03/201123:01:48‹william›Suppose <math>M</math> Is compact06/03/201123:02:07* '''chlee''' quit (timeout)06/03/201123:02:31‹william›Let <math>c</math> be such that <math>df_1 \wedge \cdots \wedge df_k |_c \neq 0</math>06/03/201123:03:52‹william›(a generic value of the system)06/03/201123:03:59‹william›Theorem:06/03/201123:04:06‹william›There is an open set <math>U</math> of <math>M</math> around <math>c</math> such that06/03/201123:04:24‹william›<math>U</math> is symplectomorphic to <math>D \times \T^n</math>06/03/201123:04:37‹william›<math>U</math> is symplectomorphic to <math>D \times T^n</math>06/03/201123:04:56‹william›where <math>D</math> is a small disc in <math>\R^n</math> and <math>T</math> is the torus, ie. <math>S^1</math>06/03/201123:05:33‹william›the symplectic form on <math>U</math> translates to <math>\sum \psi_i \wedge \phi_i</math>06/03/201123:05:55‹william›where <math>\psi</math> is standard coords for <math>D</math>, <math>\phi</math> standard circular coords for <math>T^n</math>06/03/201123:06:12‹william›the function <math>f</math> depends only on <math>D</math>,06/03/201123:06:32‹william›ie. fibres <math>f^{-1}(y)</math> are the sets <math>\{t\} \times T^n</math>06/03/201123:07:22‹william›(also can assume that <math>f^{-1}(c) = \{0\} \times T^n</math>06/03/201123:07:24‹william›)06/03/201123:08:19‹william›sorry, <math>c</math> is actually a point of <math>\R^n</math> such that <math>df_1 \wedge \cdots \wedge df_n</math> is never zero on <math>f^{-1}(c)</math>06/03/201123:08:44‹william›if this happens <math>f^{-1}(c)</math> is called a generic fibre06/03/201123:12:17‹william›Finally, the equation <math>x'(t) = X_H(x(t))</math>06/03/201123:12:38‹william›simplifies to <math>x'(t) = c</math>, where I believe <math>c</math> depends only on <math>D</math>06/03/201123:14:17‹william›Why? <math>X_H</math> is a linear combination of the <math>X_{f_i}</math>'s06/03/201123:14:42‹william›The <math>X_{f_i}</math>'s are tangent to the fibres <math>f^{-1}(y)</math>06/03/201123:18:39‹william›The flows for <math>X_{f_i}</math> are used to construct coordinates for <math>f^{-1}(c)</math>06/03/201123:18:55‹william›which means that <math>X_H</math> being a linear combination of <math>X_{f_i}</math>'s06/03/201123:19:30‹william›we get that <math>x'(t) = X_H(x(t))</math> simplies to <math>x'(t) = v</math>, <math>v</math> some constant vector06/03/201123:21:28‹william›On <math>D \times T^n</math>, <math>X_{f_i}</math> looks like <math>(d,t) \mapsto (d,t,g(d) v)</math>06/03/201123:21:36‹william›<math>v</math> a fixed vector in <math>\R^n</math>06/03/201123:22:17‹william›<math>D \times T^n \rightarrow D \times T^n \times \R^{2n}</math>06/03/201123:23:17‹william›If you fix <math>d \in D</math>, then the restriction of <math>X_{f_i}</math> to <math>\{d\} \times T^n</math> looks like <math>(d,t) \mapsto (d,t,v)</math>06/03/201123:23:33‹william›<math>v \in \R^n</math>06/03/201123:23:50‹william›In fact, <math>v</math> is the coordinate vector cooresponding to coordinate <math>\phi_i</math>06/03/201123:27:54‹william›In the non-compact case, fibres are <math>\R^k \times T^{n-k}</math>06/03/201123:28:20‹william›<math>f^{-1}(c) \iso \R^k \times T^{n-k}</math>06/03/201123:30:07‹william›Liouville-Arnold integrability06/03/201123:30:18‹william›<math>\T^n</math> are called Liouville torii06/03/201123:30:37‹william›<math>\psi</math> and <math>\phi</math> are called action-angle coords06/03/201123:31:07‹william›[http://en.wikipedia.org/wiki/Action-angle_coordinates http://en.wikipedia.org/wiki/A...dinates]06/03/201123:47:44‹william›<math>x'(t) = X_H(x(t))</math>
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"Symplectic leaves"의 두 판 사이의 차이

2012년 6월 9일 (토) 15:58 판

3/6/2011

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